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Mirrors > Home > MPE Home > Th. List > gexid | Structured version Visualization version GIF version |
Description: Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
gexcl.1 | ⊢ 𝑋 = (Base‘𝐺) |
gexcl.2 | ⊢ 𝐸 = (gEx‘𝐺) |
gexid.3 | ⊢ · = (.g‘𝐺) |
gexid.4 | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
gexid | ⊢ (𝐴 ∈ 𝑋 → (𝐸 · 𝐴) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6657 | . . . 4 ⊢ (𝐸 = 0 → (𝐸 · 𝐴) = (0 · 𝐴)) | |
2 | gexcl.1 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
3 | gexid.4 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | gexid.3 | . . . . 5 ⊢ · = (.g‘𝐺) | |
5 | 2, 3, 4 | mulg0 17546 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (0 · 𝐴) = 0 ) |
6 | 1, 5 | sylan9eqr 2678 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 = 0) → (𝐸 · 𝐴) = 0 ) |
7 | 6 | adantrr 753 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (𝐸 · 𝐴) = 0 ) |
8 | oveq1 6657 | . . . . . . 7 ⊢ (𝑦 = 𝐸 → (𝑦 · 𝑥) = (𝐸 · 𝑥)) | |
9 | 8 | eqeq1d 2624 | . . . . . 6 ⊢ (𝑦 = 𝐸 → ((𝑦 · 𝑥) = 0 ↔ (𝐸 · 𝑥) = 0 )) |
10 | 9 | ralbidv 2986 | . . . . 5 ⊢ (𝑦 = 𝐸 → (∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 )) |
11 | 10 | elrab 3363 | . . . 4 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ↔ (𝐸 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 )) |
12 | 11 | simprbi 480 | . . 3 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } → ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 ) |
13 | oveq2 6658 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 · 𝑥) = (𝐸 · 𝐴)) | |
14 | 13 | eqeq1d 2624 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐸 · 𝑥) = 0 ↔ (𝐸 · 𝐴) = 0 )) |
15 | 14 | rspcva 3307 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 ) → (𝐸 · 𝐴) = 0 ) |
16 | 12, 15 | sylan2 491 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }) → (𝐸 · 𝐴) = 0 ) |
17 | elfvex 6221 | . . . 4 ⊢ (𝐴 ∈ (Base‘𝐺) → 𝐺 ∈ V) | |
18 | 17, 2 | eleq2s 2719 | . . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐺 ∈ V) |
19 | gexcl.2 | . . . 4 ⊢ 𝐸 = (gEx‘𝐺) | |
20 | eqid 2622 | . . . 4 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } | |
21 | 2, 4, 3, 19, 20 | gexlem1 17994 | . . 3 ⊢ (𝐺 ∈ V → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 })) |
22 | 18, 21 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑋 → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 })) |
23 | 7, 16, 22 | mpjaodan 827 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝐸 · 𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℕcn 11020 Basecbs 15857 0gc0g 16100 .gcmg 17540 gExcgex 17945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-mulg 17541 df-gex 17949 |
This theorem is referenced by: gexdvdsi 17998 gexod 18001 gex1 18006 pgpfac1lem3a 18475 |
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